Question: Find the slope and y-intercept of the line that is ${\text{perpendicular}}$ to $\enspace {y = -3x + 4}\enspace$ and passes through the point ${(5, 3)}$. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$
Explanation: Lines are considered perpendicular if their slopes are negative reciprocals of each other. The slope of the blue line is ${-3}$ , and its negative reciprocal is ${\dfrac{1}{3}}$ Thus, the equation of our perpendicular line will be of the form $\enspace {y = \dfrac{1}{3}x + b}\enspace$ We can plug our point, $(5, 3)$ , into this equation to solve for ${b}$ , the y-intercept. $3 = {\dfrac{1}{3}}(5) + {b}$ $3 = \dfrac{5}{3} + {b}$ $3 - \dfrac{5}{3} = {b} = \dfrac{4}{3}$ The equation of the perpendicular line is $\enspace {y = \dfrac{1}{3}x + \dfrac{4}{3}}\enspace$. ${m = \dfrac{1}{3}, \enspace b = \dfrac{4}{3}}$